Text and diagrams © 2001 by Joseph L. Monzo
with helpful comments by Paul Erlich
(This page is accompanied with audio of the beginning of my version
of Mozart's 40th Symphony in G-minor, K. 550, tuned in a subset of 55-EDO.)
It is known that Leopold Mozart taught his son Wolfgang to use a meantone tuning, where "flats" would be higher in pitch than "sharps". There is no documentation as to exactly what this tuning was, but there are a few clues.
Paul Erlich quoted in Yahoo Tuning List message 24125 (Thu May 31, 2001 9:03 pm) from Chesnut 1977 as follows:
Leopold Mozart refers to Tosi in general terms as an authoritative source in a letter to Wolfgang from Salzburg dated June 11, 1778. Tosi, in 1723, considered the correct tuning system to be what we would today call a form of regular meantone temperament ... according to Tosi, the large diatonic half step is theoretically equal to five ninths of a whole step, and the small chromatic half step is theoretically four-ninths of a whole step. Tosi thereby divides the octave into fifty-five equal parts. This is equivalent to tempering the perfect fifth by approximately one-sixth of a '[syntonic-] comma,' ...
Leopold Mozart, in his violin method of 1756 -- which happens to be the year of Wolfgang's birth -- also describes what we have called 'extended regular meantone temperament' as the correct intonation for the violin; he tells us that keyboard instruments of his time were played with some form of tempered [i.e., well-tempered] tuning, but that in the "right ratio" [i.e., meantone] tuning that he recommends for the violin, flats are higher by a comma than enharmonically equivalent sharps. It can be shown that for whichever of the standard commas we choose, the perfect fifths in Leopold Mozart's system were theoretically flattened by about one-sixth of that comma . . . Leopold Mozart wrote down a couple of scales specifically intended for practice in intonation, one leading through the flats, the other through the sharps. In practicing these scales, the student is supposed to learn to distinguish between the large diatonic half steps and the small chromatic half steps. It is important to emphasize that these scales are not abstractions but exercises to be mastered . . .
Thus it is apparent that for instruments other than keyboards (which both Mozart and his predecessors tuned in various well-temperaments, typically resembling 1/6-comma meantone for the "natural" notes, and pythagorean for the "chromatic" keys), Mozart's tuning would be based on 55-EDO, or something very closely approximating it. It would have the following sizes for the basic intervals:
ratio Semitones cents
1 55-EDO degree 2(1/55) 12/55 21 9/11
smaller (chromatic) semitone 2(4/55) 48/55 87 3/11
larger (diatonic) semitone 2(5/55) 1 1/11 109 1/11
whole tone 2(9/55) 1 53/55 196 4/11
55-EDO renders two different-sized semitones but only one size whole-tone; thus, it is a meantone. (Compare with the 50-EDO tuning recommended as a very close approximation to an "optimal meantone" by Woolhouse in 1835; and contrast with 53-EDO, which is so close to both pythagorean and 5-limit JI that it gives two different-sized whole-tones.)
On this webpage I will first explore a 12-tone subset of 55-EDO, as I had mistakenly thought that it was the most likely intended tuning for Mozart's music. I have since learned that Mozart actually intended an interesting 20-tone subset of 55-EDO, and I give an update below on that. I decided to keep the info about the 12-tone subset because many people with ordinary Halberstadt keyboards may wish to explore that subset of 55-EDO, as it still gives a nice meantone system.
Note that for non-keyboard instruments, Mozart's actual conception of this tuning extended in a meantone cycle from Ebb to A#, with Cb omitted, for a total of 20 notes, thus giving two different pitches for the pairs of "sharp and "flat" notes (which are each one "black key" on the Halberstadt keyboard), and alternates for several of the "natural" notes too. So there is no evidence that Mozart ever actually used or advocated a 12-tone subset of this tuning (which is what I present here).
The mapping uses "D" as the symmetrical pitch center, and renders the 12-tone subset as a meantone "cycle of 5ths" from Eb to G#, with the "wolf" appearing between those two pitches.
2(32/55) = ~698.181818... cents,is equivalent to that of ~0.175445544-comma meantone. To describe that in terms of low-integer fraction-of-a-comma meantones, that's
a little less than 1/5 of a cent (just slightly more than 3/16, extremely close to 7/37, and almost exactly 10/53) narrower than the 1/6-comma meantone "5th" = ~698.3706193 cents,
even closer (less than 1/7 of a cent wider) to the 2/11-comma meantone "5th" = ~698.0447664 cents,
much closer (~1/46-cent wider) to the 3/17-comma meantone "5th" = ~698.1597733 cents,
closer still (less than 1/100-cent narrower) to the 7/40-comma meantone "5th" = ~698.1914002 cents,
and almost exactly (~1/6692-cent wider than) the 10/57-comma meantone "5th" = ~698.1819676 cents.
The amount of tempering in 1/6-comma meantone is:
(81/80)(1/6) = (2-4 * 34 * 5-1)(1/6) = 2(-2/3) * 3(2/3) * 5(-1/6)
= ~3.584381599 cents = ~3 & 3/5 cents.
So the 1/6-comma meantone "5th" is (3/2) / ((81/80)(1/6)). Using vector addition, that's:
2^ 3^ 5^ | -1 1 0 | = 3/2 - | -2/3 2/3 -1/6 | = (81/80)(1/6) ----------------------- | -1/3 1/3 1/6 | = 1/6-comma meantone "5th" = ~698.3706193 cents.
Using vector addition again to compare the 1/6-comma meantone "5th" with the 55-EDO "5th", we get:
2^ 3^ 5^ | -1/3 1/3 1/6 | = 1/6-comma meantone "5th" - | 32/55 0 0 | = 55-EDO "5th" ------------------------- | -151/165 1/3 1/6 | = 1/6-comma "5th" "-" 55-EDO "5th" = ~0.188801084 cent = ~1/5 or ~10/53 cent, as stated above.
Below I present a side-by-side comparison of 55-EDO and 1/6-comma meantone.
First, the 12-out-of-55-EDO subset and the 12-tone 1/6-comma meantone presented as cycles of "5ths" from Eb to G#:
Next, both of the scales presented as 12-tone chromatic scales, with their cycle-of-"5ths" and Semitone values, the Semitones and 55-EDO degrees between notes in the scale, and the 55-EDO degree representing the scale notes:
And last, the interval matrices for both tunings, showing the 55-EDO degrees for the 12-o-o-55 subset and the "cycle-of-5ths" value for the 1/6-comma meantone, with the Semitone values of all possible intervals given in the body of the table. Some important intervals are color-coded for easy recognition:
Here is the above table reproduced with fractional cents values instead of decimal Semitones:
Note also the following comment by Paul Erlich (from a Yahoo Tuning List post), echoing what Chesnut wrote at the end of his article:
There is no real evidence that Mozart intended 1/6- comma meantone rather than 1/5-comma or some other meantone system. It wouldn't make much sense for one to be that specific in regard to Mozart anyway, since he clearly assumes enharmonic equivalence even in a few passages for strings unaccompanied by keyboard, which violates his known violin teaching altogether.
Joe Monzo does not necessarily agree with this, believing that while Mozart in his keyboard music obviously would have to intend his enharmonically-equivalent pairs of notes to be exactly the same pitch, in non-keyboard passages he may in fact have intended the commatic shifts of two different pitches to be played when he notated "enharmonically-equivalent" pairs of notes tied together.
Chesnut, John Hind. 1977.
"Mozart's teaching of intonation",
Journal of the American Musicological Society
vol. 30 no. 2 [summer], pp. 254-271.
section above updated: 2001.06.03, 2001.07.08
Here is a graph of a 56-tone cycle of 1/6-comma meantone, centered on "C" as the reference (= generator 0):
As seen above, extending the cycle beyond 12 pitches results in pairs of pitches separated by approximately a comma. Upon passing beyond the -27..+27 cycle, note that the pitches represented as Cbbbb and Fxx are closer than that:
Fxx = ( (3/2)27 / ( (81/80)(27/5) ) ) / 215 = 2-(144/6) * 3(54/6) * 5(27/6) = ~856.0067202 cents Cbbbb = ( (3/2)-28 / ( (81/80)(-28/5) ) ) / 212 = 2(158/6) * 3-(56/6) * 5-(28/6) = ~845.6226606 cents 2-(144/6) * 3(54/6) * 5(27/6) Fxx - 2(158/6) * 3-(56/6) * 5-(28/6) Cbbbb ---------------------------- 2-(302/6) * 3(110/6) * 5(55/6) = ~10.38405963 (= 10 ~3/8) cents. 10.38405963 / 55 = 0.188801084 --> the difference between the "5ths" of 1/6-comma meantone and 55-edo, which was noted above.
Thus, limiting the meantone cycle to 55 pitches and distributing this difference equally among them, results in 55-EDO. If we call the 21 9/11 cent step-size of 55-EDO a "comma", then the "chromatic-semitone" = 4 commas, the "diatonic-semitone" = 5 commas, and the "whole-tone" = 9 commas, exactly as Mozart taught his students.
Other EDOs which approximate 1/6-comma meantone more
closely than 55-EDO are 67- and 122-EDO:
2(39/67) = ~698.5074627 cents
2(71/122) = ~698.3606557 cents
(This section added 2001.11.08)
I'm adding a new section here which explores Mozart's actual intended tuning. Knowledge of this tuning is based on notes written down by Thomas Atwood, who studied with Mozart during the mid-1780s. My source for this is the Chesnut article cited above.
Mozart used the same process as given above, but extended it to a cycle of 21 meantone "5ths", an elegantly symmetrical system from implied 3-10 = Ebb to implied 310 = A#. He then destroyed the symmetry of the scale by limiting himself to 20 notes and eliminating the implied 3-7 = Cb.
It's curious to me that Mozart decided to do this, rather than keep the symmetrical tuning. It's interesting that the very note he leaves out is the one which would become the most characteristic feature of the later development (just after Mozart's death) of "expressive" pythagorean-based intonation: the "raised" (sharpened or narrowed) "leading-tone" [again, see the Chesnut article for details].
Chesnut notes that Tosi advocated 55-EDO, and that one degree or "step" in this tuning (i.e., the ratio 2(1/55) = 21 & 9/11 cents) is so close to both the syntonic and pythagorean commas that it could be called a "Tosi comma". There are exactly 4 & 7/12 of these "Tosi commas" in one 12-EDO Semitone.
Below is the mapping of the basic prime intervals in 55-EDO:
- "per" = number of instances of the period of equivalence needed for "correction" - in this case, "octaves" - to put the generated interval into the proper register to represent the basic prime interval.
- "gen" = number of 2(32/55) generators from 0. In this temperament, the generator is considered to be the "5th".
This notation thus gives the 55-EDO analogue of a vector notation using 2 (the "octave") and 3/2 (the "5th") as factors.
32/55-EDO mapping prime (per, gen) ~cents error ~ % error 2 ( 1, 0 ) 0 0 3 ( 1, 1 ) -3.773182684 -0.198384435 5 ( 0, 4 ) 6.413558862 0.230180788 7 (-3, 10) 12.99227535 0.385661821 11 (-7, 18) 15.95478491 0.384330594
(For a detailed explanation of this type of matrix notation, see Graham Breed's matrix webpage. Graham would define 1/6-comma meantone as follows:
factors: 2 3 5 implied ratio ( 1 0 0 ) (1200) 2:1 H' = ( 2/3 1/3 1/6)H = (1898) cents ~3:1 (-4/3 4/3 2/3) (2793) ~5:1
And 55-EDO as follows:
)factors: 2 3 5 implied ratio ( 1 0 0) (1200) 2:1 H' = ( 87/55 0 0)H = (1898) cents ~3:1 (128/55 0 0) (2793) ~5:1
Note that the generator which represents 11 lies outside the 21-tone cycle given in the above graph, thus, this particular scale only provides three approximate 11:8s, between the pairs of notes (expressed as generators and pitches where C = 0): (-10, 8) = Ebb:G#, (-9, 9) = Bbb:D#, and (-8, 10) = Fb:A#. This can be seen in the interval matrix further below.
Below is a lattice illustrating the relationship of 1/6-comma meantone with the 5-limit JI pitch-classes it implies. (This particular example illustrates a symmetrical 27-tone chain of 1/6-comma meantone "5th"s; it could be extended in either direction.)
(lattice added 2001.12.3)
Yahoo Tuning Group, Message 45865 From:
Date: Sat Jul 26, 2003 3:56 am Subject: smoking gun: W.A. Mozart's writings on intonation i wrote: http://groups.yahoo.com/group/tuning/message/44784 > From: "monz" > Date: Tue Jun 17, 2003 5:54 pm > Subject: Re: [tuning] Re: Mozart 31-equal?? > > > > > ... right now i'm busy tracking down a copy of > the Attwood studies to see what W.A. Mozart himself > wrote about intonation. and now i've gotten it, and can share what i found. (thank the gods for interlibrary loan!) basically there's nothing new that i didn't already glean from Chesnut's article, and which i've already included on my 55edo webpage: http://sonic-arts.org/monzo/55edo/55edo.htm [old link, now redirects here] The largest set of pitches actually written in Mozart's handwriting is a 21-tone meantone chain of 5ths Ebb ... A#, but without Cb : Ebb Bbb Fb [Cb] Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# Attwood apparently understood Italian better than German, because Mozart wrote in Italian for his benefit. It is clearly stated several times in both of their handwritings that the 8ve contains 5 tones and 2 large semitones. The notes begin with ascending and descending (one 8ve) C-major and A-melodic-minor scales, in Mozart's handwriting, with notes explaining where the semitones occur. then (p I/1 and I/2) Mozart wrote the first note of several other major scales and their relative minors, and Attwood filled in all the other notes correctly, except for the leading-tone of G#-minor, which he wrote as F# ascending (corrected by Mozart to Fx) and F-natural descending (corrected by Mozart to F#). the list of major scales is, in Mozart's order, C F Bb Eb Ab Db Gb B E A. the relative minors are thus A D G C F Bb Eb G# C# F#. the 21-note meantone chain of 5ths which contains all the notes included in these scales is thus: Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx page I/4, entirely in Mozart's handwriting except for one sentence by Attwood, is a list of large and small semitones and then a table of intervals arranged in 8ve-complementary pairs (and continued on page I/5). Mozart's list of semitones is: large semitones: C-Db, C#-D, D-Eb, D#-E, E-F, F-Gb, F#-G small semitones: C-C#, Db-D, D-D#, Eb-E, E-E#, F-F#, Gb-G (the complete text is quoted below) page I/10 contains Attwood's attempt at making a table of all the meantone intervals from a reference pitch of A#, and it is remarkable for the number of errors in what is otherwise a rather well-done set of studies -- i have marked them all and indicated the corrections, which Mozart didn't even bother to do. Mozart instead preferred to write a new table of intervals on the next page, again using a reference pitch of C, as on pages I/4-5, giving Attwood only one of each pair of 8ve-complementary intervals and having Attwood write the complement. Attwood finally got all of information for semitone sizes correct on page I/25, and the interval sizes on pages I/26 and I/27, none of which i have included as they simply recap what Mozart wrote. however, he does note that in the cases of Ebb, Fb, and Bbb, "these tones the Harpsichord has not, but all other Instruments have ---". the numerous other pages of the book contain harmony and counterpoint exercises. now, it is true that the relative sizes of the two different semitones it is not stated anywhere in this book. however, we know that Mozart's father Leopold taught that the flats were a comma higher than the sharps, which indicates 55edo or another meantone similar to it. ------- begin pages quoted from Attwood's notes ---------- * I/1, p. 3 * Nella scala maggiore il 1mo mezzo tuono Ã¨ dalla terza alla quarta. ed il 2do dalla 7ma all' octava. il lmo dall' octava all 7ma ed il 2do dalla quarta alla terza. [ (Monzo's English translation:) In the major-scale the first semitone is between the 3rd and 4th, and the second [semitone is] between the 7th and 8ve. [descending:] The first [semitone is] between the 8ve and 7th and the second [semitone is] between the 4th and 3rd. ] * I/4, p. 6 * [see I/10, p. 10] [in Attwood's hand:] A tone ought to have a little tone & half a great one [Attwood mangled this: he meant to say "A tone ought to have one little half-tone and one great one".] [all music examples on this page and the next in Mozart's hand:] Mezzi tuoni grandi [large semitones] C-Db, C#-D, D-Eb, D#-E, E-F, F-Gb, F#-G mezzi tuoni piccoli. [small semitones] C-c#, Db-D, D-D#, Eb-E, E-E#, F-F#, Gb-G [the following list gives all the basic intervals, with an example and then another example giving the inversion or octave-complement] unisono [unison, i.e. prime] C-C mezzo tuono piccolo. [small semitone] unisono superfluo [augmented-prime] C-C# 8tava diminiuta [diminished-8ve] C#-C mezzo tuono grande. [large semitone] seconda minore [minor-2nd] C-Db 7ma maggiore [major-7th] Db-C un tuono. [tone, i.e. whole-tone] seconda maggiore [major-2nd] C-D 7ma minore [minor-7th] D-C 1 tuono ed un mezzo tuono piccolo. [1 whole-tone + 1 small semitone] seconda superflua [augmented-2nd] C-D# [i.e. C <w-t> D <s.s-t> D# or C <s.s-t> C# <w-t> D#] 7ma diminuta [diminished-7th] D#-C 1 tuono, ed un mezzo tuono gr. [1 whole-tone + 1 large semitone] terza min: [minor-3rd] C-Eb [i.e. C <w-t> D <l.s-t> Eb or C <l.s-t> Db <w-t> Eb] 6ta mag: [major-6th] Eb-C 2 tuoni [2 whole-tones] terza mag: [major-3rd] C-E [i.e. C <w-t> D <w-t> E] 6ta min: [minor-6th] E-C 1 tuono, e 2 mezzi tuoni gr: [1 whole-tone + 2 large semitones] quarta diminuta [diminished-4th] C-Fb [i.e. C <l.s-t> Db <w-t> Eb <l.s-t> Fb] 5ta superflua [augmented-5th] Fb-C 2 tuoni, ed un semitonio gr: [2 whole-tones + 1 large semitone] quarta minore ["minor-4th", i.e. perfect-4th] C-F [i.e. C <w-t> D <w-t> e <l.s-t> F] 5ta reale [perfect-5th] F-C 3 tuoni [3 whole-tones] quarta mag: ["major-4th", i.e. augmented-4th] C-F# [i.e. C <w-t> D <w-t> E <w-t> F#] 5ta falsa: ["false-5th", i.e. diminished-5th] F#-C [actually an incorrect example] 2 tuoni, e 2 mezzi tuoni gr: [2 whole-tones + 2 large semitones] quinta falsa ["false-5th", i.e. diminished-5th] C-Gb [i.e. C <l.s-t> Db <w-t> Eb <w-t> F <l.s-t> Gb] 4ta mag: ["major-4th", i.e. augmented-4th] Gb-C 3 tuoni, ed un semit: gr: [3 whole-tones + 1 large semitone] quinta vera o 5ta reale [perfect-5th] C-G [i.e. C <w-t> D <w-t> E <l.s-t> F <w-t> G] 4ta minore ["minor-4th", really perfect-4th] G-C 3 tuoni, un semit: gr. ed uno pic: [3 whole-tones + 1 large semitone + 1 small semitone] quinta superflua [augmented-5th] C-G# [C <w-t> D <w-t> E <w-t> F# <l.s-t> G <s.s-t> G#] 4ta diminuta: [diminished-4th] G#-C * I/5, p. 7 * ... 3 tuoni, e 2 semit: gr: [3 whole-tones + 2 large semitones] sesta minore [minor-6th] C-Ab [i.e. C <w-t> D <l.s-t> Eb <w-t> F <w-t> G <l.s-t> Ab] terza mag: [major-3rd] Ab-C 4 tuoni, ed un semit: gr: [4 whole-tones + 1 large semitone] sesta mag: [major-6th] C-A [i.e. C <w-t> D <w-t> E <l.s-t> F <w-t> G <w-t> A] terza min: [minor-3rd] A-C 4 tuoni, un semit: gr: ed uno pic: [4 whole-tones + 1 large semitone + 1 small semitone] sesta superflua [augmented-6th] C-A# [i.e. C <w-t> D <w-t> E <l.s-t> F <w-t> G <w-t> A <s.s-t> A#] terza diminuta: [diminished-3rd] A#-C 4 tuoni, e 2 semit: gr: [4 whole-tones + 2 large semitones] septima min: [minor-7th] C-Bb [i.e. C <w-t> D <w-t> E <l.s-t> F <w-t> G <w-t> A <l.s-t> Bb] 2da mag: [major-2nd] Bb-C 5 tuoni, ed un semit: gr: [5 whole-tones + 1 large semitone] septima mag: [major-7th] C-B [i.e. C <w-t> D <w-t> E <l.s-t> F <w-t> G <w-t> A <w-t> B] 2da min: [minor-2nd] B-C 5 tuoni, e 2 semit: grandi [5 whole-tones + 2 large semitones] 8tava [8ve] C-C (high) [i.e. C <w-t> D <w-t> E <l.s-t> F <w-t> G <w-t> A <w-t> B <l.s-t> C] unisono: [unison, i.e. prime] C-C [in Attwood's hand:] An Octave must have 5 tones & two Great half tones. * I/6, p. 8 * C C#,Db D D#,Eb E E#,F F#,Gb G G#,Ab A A#,Bb B B# c [meantone chain: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B#] * I/10, P. 10 * [see I/4, p. 6] [entire page in Attwood's hand, and filled with errors:] Unis: Mez. t Gr A#-B, B-C, B#-C#, C#-D, Cx-D#, D#-E, Dx-E#, E#-F#, Ex-Fx, Fx-G# Mez. t p: A#-Ax, B-B#, C-C#, C#-Cx, D-D#, D#-Dx, E-E#, E#-Ex, F#-Fx, G-G# [meantone chain: C G D [A] E B F# C# G# D# A# E# B# Fx Cx [Gx] Dx Ax Ex] Antonio Sdadler [?} Unis A#-A# 8tava A#(low)-A# mez t pic. unis sup: A#-Ax 8ta diminuta Ax-A# mez Gr: 2do Minore A#-B 7ma Mag: Bb-A# [incorrect: should be B-A#] 1 t. 2da Mag: A#-B# 7ma min. B-A# [incorrect: should be B#-A#] 1 t e un mez p: 2 sup: A#-Bx 7ma dim: B#-A# [incorrect: should be Bx-A#] 2 mag gr 3za dimi: A#-C [Cbb crossed out] 6ta Super Cbb-A# [incorrect: should be C-A#] 1 t e mag gr 3za Minore A#-C# 6ta Mag. Cb-A# [incorrect: should be C#-A#] 2 t p: 3 Mag: A#-Cx 6ta Min C-A# [incorrect: should be Cx-A#] 1 t: e dui mez. gr: 4ta dimi A#-D [Dbb crossed out] 5ta Super Dbb-A# [incorrect: should be D-A#] 2 t e mez Gr 4 Minore A#-D# [Db crossed out] 5ta Mag Db-A# [incorrect: should be D#-A#] why can't the fourth have a Superflua [Attwood's marginal note] 3 t: 4ta Mag: A#-D [natural sign crossed out] [incorrect: should be A#-Dx] 5 Mag. D-A# [incorrect: should be Dx-A#] 2 t e 2 mez Gr 5ta falsa A#-Ebb [incorrect: should be A#-E] 4 Mag Ebb-A# [incorrect: should be E-A#] 3 t e mez Gr 5ta vera A#-Eb [incorrect: should be A#-E#] 4 Minore Eb-A# [incorrect: should be E#-A#] 3 t e mag Gr e mez p: 5ta Super A#-E [incorrect: should be A#-Ex] 4ta dimi E-A# [incorrect: should be Ex-A#] 3 t e 2 mag Gr 6ta minore A#-Fb [incorrect: should be A#-F#] 3za Mag: Fb-A# [incorrect: should be F#-A#] 4 t e mez Gr 6ta Mag. A#-F [incorrect: should be A#-Fx] 3za Minore F-A# [incorrect: should be Fx-A#] 4 t e 2 Gr mag p: [incorrect: should be 4 t e mez Gr: e mez p:] 6ta Super A#-F# [incorrect: should be A#-FX#] 3za dimi F#-A# [incorrect: should be Fx#-A#] 4 t e mez Gr: [incorrect: should be 3 t e 3 mez Gr:] 7ma dim: A#-Gbb [incorrect: should be A#-G] 2 Super Gbb-A# [incorrect: should be G-A#] 4 t e dui Mez Gr. 7 minore A#-Gb [incorrect: should be A#-G#] 2 Mag Gb-A# [incorrect: should be G#-A#] 5ta e mez Gr 7ma Mag. A#-G [incorrect: should be A#-Gx] 2 Super G-A# [incorrect: should be Gx-A#] 5ta e 2 mez Gr 8tava A#-G# [incorrect: should be A#-A#(high)] unis. G#-A# [incorrect: should be A#-A#] [meantone chain for Attwood's incorrect notes: Cbb Gbb Dbb [Abb] Ebb [Bbb] Fb Cb Gb Db [Ab] Eb Bb F C G D [A] E B F# C# G# D# A# [E#] B# [Fx] Cx [Gx] [Dx] Ax [Ex] Bx - clearly incorrect, no corrections given by Mozart -- Mozart wrote out a new example on next page.] [the correct meantone chain for the intervals given by Attwood from A# is: C G D [A] E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx Fx#] * I/11, p. 11 * [all top intervals in Mozart's hand, all bottom intervals in Attwood's:] unis: C-C [Mozart] otava C(low)-C [Attwood, etc.] unis: superfl: C-C# 8tava diminuta C#-C 2da min: C-Db 7 mag: Db-C 2da mag: C-D 7ma minore D-C 2da sup: C-D# 7ma diminuta D#-C 3: dim: C-Ebb 6ta super: Ebb-C 3: min: C-Eb 6ta mag: Eb-C 3 mag: C-E 6ta minore E-C 4: dim: C-Fb 5ta super: Fb-C 4ta min: C-F 5ta vera F-C 4: mag: C-F# 5ta falsa F#-C 5: falsa C-Gb 4 mag: Gb-C 5: vera C-G 4 minore G-C 5: sup: C-G# 4 diminuta G#-C 6: min: C-Ab 3za mag: Ab-C 6: mag: C-A 3 minore A-C 6: sup: C-A# 3za dim: A#-C 7ma dim: C-Bbb 2da super Bbb-C 7: min: C-Bb 2 Mag: Bb-C 7: mag: C-B 2 minore B-C Octava C-C (high) unis C-C [i.e., thus this is the same as I/4-5, except that it also adds diminished-3rd/superfluous-6th and diminished-7th/superfluous-2nd pairs.] [meantone chain: Ebb Bbb Fb [Cb] Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A#] [on I/26,p.28 and I/27,p.29, Attwood rewrites exactly this same diagram, with English names for the intervals.] ----- end pages quoted from Attwood's notes -------- REFERENCE --------- Heartz, Daniel; Mann, Alfred; Oldman, Cecil B.; Hertzmann, Erich. 1965. _Thomas Attwoods Theorie- und Kompositionsstudien bei Mozart_. Wolfgang Amadeus Mozart: Neue Ausgabe sÃ¤mtlicher Werke, Ser. X, Werkgruppe 30, Bd. 1. Kassel. -monz
Table of 1/6-comma meantone reference pitches in Hz, in relation to 1/1 = A 440 Hz, for the octave between the tritones surrounding A-440, for all of the 31 notes normally used in the Western musical repertoire:
Below is a similar table, again for 1/6-comma meantone and again in relation to A-440, but bounded instead by the B# below middle-C at the bottom and Cb above A-440 at the top, arranged in order of pitch-height, and including a graph of the pitches: